Hopf bifurcation analysis in a synaptically coupled FHN neuron model with delays

Fan, Dejun; Hong, Li

doi:10.1016/j.cnsns.2009.07.025

Cited by 40 publications

(17 citation statements)

References 20 publications

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“…Clearly, the linear part of system (1.1) at the trivial equilibrium is the same as (2.1) in [15] while C 1 = C 2 , τ 1 = τ 2 and ignoring the differences of the notation.…”

Section: Lemma 23 Equation (11) Has a Triple Zero Eigenvalue If Andmentioning

confidence: 97%

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Hopf and Bogdanov–Takens bifurcations in a coupled FitzHugh–Nagumo neural system with delay

Jiang

2010

Nonlinear Dyn

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In this paper, Hopf bifurcation and Bogdanov-Takens bifurcation with codimension 2 in a coupled FitzHugh-Nagumo neural system with gap junction are investigated. At first, a general bifurcation diagram on the plane of coupling strength and delay is derived. Then, explicit algorithms due to Hassard and Faria are applied to determine the normal forms of Hopf and Bogdanov-Takens bifurcations, respectively. Next, we analyze the codimension-2 unfolding for Bogdanov-Takens bifurcation, and give complete bifurcation diagrams and phase portraits. Furthermore, we also consider the spatio-temporal patterns of bifurcating periodic solutions by using the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups. By the results of theoretical analysis, we obtain that the values of coupled strength, which make the transmission and received signals be synchronous and anti-phase, are opposite. And universal unfolding of BogdanovTakens bifurcation indicates that the neuron signals can transit between resting and spiking.

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“…Clearly, the linear part of system (1.1) at the trivial equilibrium is the same as (2.1) in [15] while C 1 = C 2 , τ 1 = τ 2 and ignoring the differences of the notation.…”

Section: Lemma 23 Equation (11) Has a Triple Zero Eigenvalue If Andmentioning

confidence: 97%

“…Some results on bifurcations for the system (1.1) have been derived. [15] discussed the stability and Hopf bifurcation of double delay coupled FHN system with different coupled strength; [16] gave the results on fold bifurcation of (1.1) with f (x) = tanh(x); [17] studied Bautin bifurcation of completely synchronous FHN neurons.…”

Section: Introductionmentioning

confidence: 99%

Hopf and Bogdanov–Takens bifurcations in a coupled FitzHugh–Nagumo neural system with delay

Jiang

2010

Nonlinear Dyn

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“…Motivated by Lemma 2.3 of the work of Ruan and Wei [53], Lemma 2.4 of the work of Li and Wei [52], Lemma 2.5 of the paper of Hu and Huang [54], and Theorem 2.1 of Fan and Hong [36], we obtain following conclusions.…”

Section: ((R-h) Hypothesis) (I) If One Of the Followings Holds: (A) Smentioning

confidence: 99%

“…However, only codimension-1 bifurcations are discussed in these papers. Fan and Hong [36], and also Xu et al [37], considered the stability and Hopf bifurcation of double delay coupled FHN system with different coupling strength. Yao and Tu [38] discussed the combined effect of coupling strength and multiple delays on the stability of the rest point, and they obtained stability switches in the coupled FHN neural system with multiple delays.…”

Section: Introductionmentioning

confidence: 99%

Bifurcation structure of two coupled FHN neurons with delay

Tehrani

Razvan

2015

Mathematical Biosciences

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This paper presents an investigation of the dynamics of two coupled non-identical FitzHugh-Nagumo neurons with delayed synaptic connection. We consider coupling strength and time delay as bifurcation parameters, and try to classify all possible dynamics which is fairly rich. The neural system exhibits a unique rest point or three ones for the different values of coupling strength by employing the pitchfork bifurcation of non-trivial rest point. The asymptotic stability and possible Hopf bifurcations of the trivial rest point are studied by analyzing the corresponding characteristic equation. Homoclinic, fold, and pitchfork bifurcations of limit cycles are found. The delay-dependent stability regions are illustrated in the parameter plane, through which the double-Hopf bifurcation points can be obtained from the intersection points of two branches of Hopf bifurcation. The dynamical behavior of the system may exhibit one, two, or three different periodic solutions due to pitchfork cycle and torus bifurcations (Neimark-Sacker bifurcation in the Poincare map of a limit cycle), of which detection was impossible without exact and systematic dynamical study. In addition, Hopf, double-Hopf, and torus bifurcations of the non trivial rest points are found. Bifurcation diagrams are obtained numerically or analytically from the mathematical model and the parameter regions of different behaviors are clarified.

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“…In science and engineering, many complex networks, including coupled biological systems, neural networks, social interacting species, chemical systems, etc., could be described by control systems on networks (CSNs). In the last decade, CSNs have been discussed ardently [1][2][3][4][5][6]. In general, most applications of CSNs are built upon the stability of the equilibrium point, because the stability of these systems are in the designs and applications of complex networks.…”

Section: Introductionmentioning

confidence: 99%

A model of feedback control system on network and its stability analysis

Gao

et al. 2013

Communications in Nonlinear Science and Numerical Simulation

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Hopf bifurcation analysis in a synaptically coupled FHN neuron model with delays

Cited by 40 publications

References 20 publications

Hopf and Bogdanov–Takens bifurcations in a coupled FitzHugh–Nagumo neural system with delay

Hopf and Bogdanov–Takens bifurcations in a coupled FitzHugh–Nagumo neural system with delay

Bifurcation structure of two coupled FHN neurons with delay

A model of feedback control system on network and its stability analysis

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